A Faster Approach for Robust Set Stabilization of Boolean Control Networks

This paper presents new methods on robust set invariance, set stabilizability, and set stabilization of Boolean control networks (BCNs) subject to arbitrary disturbance inputs. We approach these problems from a novel algorithmic perspective and develop more efficient algorithms that scale to moderately large networks. The predecessor concept in graph theory is first generalized to a robust predecessor (RP) of a state set to handle nondeterministic state transitions. A direct connection between RPs and a robust control invariant set (RCIS) is established. After that, a simple recursive elimination algorithm is developed to identify the largest RCIS quickly. A faster method is then proposed to check robust set stabilizability and attain time-optimal robust set stabilization. Compared with existing methods, our algorithms feature lower time complexity and reduce the running time dramatically, as demonstrated by case studies for two biological networks.

1 Introduction

A Boolean network (BN) is a discrete-time logical system model initially proposed by Kauffman to describe gene regulatory networks (GRNs) [1]. In a BN, the state of a node is binary and updated via Boolean interaction with each other. When external binary inputs are involved, we get a Boolean control network (BCN). The control-theoretical study on BCNs is booming in recent years mainly thanks to the emergence of a novel mathematical tool called the semi-tensor product (STP) of matrices [2], which provides a convenient algebraic state-space representation (ASSR) of BCNs and thus enables systematic analysis of BCNs with algebraic means. In this study, we investigate the robust set stabilization problem of a BCN based on its ASSR.